to help with a down payment on a home, dan is going to invest. assuming an interest rate of 1.62% compounded…

to help with a down payment on a home, dan is going to invest. assuming an interest rate of 1.62% compounded quarterly, how much would he have to invest to have $39,400 after 6 years? do not round any intermediate computations, and round your final answer to the nearest dollar. if necessary, refer to the list of financial formulas.

to help with a down payment on a home, dan is going to invest. assuming an interest rate of 1.62% compounded quarterly, how much would he have to invest to have $39,400 after 6 years? do not round any intermediate computations, and round your final answer to the nearest dollar. if necessary, refer to the list of financial formulas.

Answer

Explanation:

Step1: Identify the compound - interest formula

The compound - interest formula is (A = P(1+\frac{r}{n})^{nt}), where (A) is the final amount ((A = 39400)), (P) is the principal amount (the initial investment we want to find), (r) is the annual interest rate (as a decimal, (r=0.0162)), (n) is the number of times interest is compounded per year ((n = 4) for quarterly compounding), and (t) is the number of years ((t = 6)). We need to solve for (P), so we can rewrite the formula as (P=\frac{A}{(1 +\frac{r}{n})^{nt}}).

Step2: Substitute the values into the formula

Substitute (A = 39400), (r=0.0162), (n = 4), and (t = 6) into the formula for (P): First, calculate (\frac{r}{n}=\frac{0.0162}{4}=0.00405) and (nt=4\times6 = 24). Then ((1+\frac{r}{n})^{nt}=(1 + 0.00405)^{24}). Using the formula (a^b=e^{b\ln(a)}), we have ((1 + 0.00405)^{24}=e^{24\ln(1.00405)}). (\ln(1.00405)\approx0.00404), so (24\ln(1.00405)\approx24\times0.00404 = 0.09696). (e^{0.09696}\approx1.1017). Alternatively, using a calculator directly: ((1.00405)^{24}\approx1.1017). Now, (P=\frac{39400}{(1.00405)^{24}}). (P=\frac{39400}{1.1017}).

Step3: Calculate the value of (P)

(P=\frac{39400}{1.1017}\approx35763).

Answer:

(35763)